Mathematics / advanced
Tools for modern theory
Develop the analytic and geometric tools behind modern physics and machine learning.
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Module 01 advanced
Tools for modern theory
Develop the analytic and geometric tools behind modern physics and machine learning.
- 01
Convergence and completeness
Decide what an approximation approaches, in which distance, and which operations survive the limit.
24 min - 02
Measure and integration
Build integrals that handle limits, probability, and functions defined only almost everywhere.
25 min - 03
Fourier analysis
Translate between position and frequency while keeping normalization, convergence, and boundaries explicit.
26 min - 04
Distributions and Green functions
Replace pointwise singularities with meaningful actions on test functions and construct responses to localized sources.
28 min - 05
Differential equations and well-posed evolution
Connect local laws, initial and boundary data, stability, and the limits of numerical time stepping.
27 min - 06
Complex analysis and contour reasoning
Use analyticity, singularities, and contour topology to evaluate integrals with defensible assumptions.
27 min - 07
Tensors and manifolds
Describe geometric objects independently of coordinates and identify what their changing components mean.
29 min - 08
Lie groups and Lie algebras
Relate continuous symmetries to infinitesimal generators without losing global topology or representation data.
29 min - 09
Functional analysis and operator domains
Treat whole functions as vectors and learn why infinite-dimensional operators need more than matrix intuition.
30 min - 10
Stochastic processes and noisy evolution
Move from one random variable to correlated histories, Markov dynamics, Brownian motion, and an honest introduction to Itô calculus.
30 min